Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation

Abstract

This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{array}{rl}& y_t-\Delta a(y)=mu+f\text{ on }Q\\ & y(0)=y_0,y{\mid }_{\Sigma }=0\end{array}$$ on the $n+1$-dimensional cylinder $Q=\Omega \times (0,T)$ with lateral boundary $\Sigma =\partial \Omega \times (0,T)$. The exact controllability in finite time is proved when $\Vert y_0-y_s{\Vert }_{W_0^{1,n}(\Omega )\cap C(\overline{\Omega })}$ is sufficiently small, $n>1$, for every stationary solution $y_s$ such that $a(y_s)\in W^{2,q}(\Omega )$, where $q>n$. It is assumed that $\Omega$ is a bounded open set with $C^2$ boundary and that $a\in C^2(\mathbb{R})$, $a^{\prime }>0$.

arXiv commentThis is an old paper, submitted to, but not accepted by, SICON in 2007--2008. This version of the paper was last edited on 11/29/2008. At some point, I may resume work on this topic and resubmit this paper as well